Stochastic Modeling

ISBN-13: 9783319500379 / Angielski / Miękka / 2017 / 303 str.

Nicolas Lanchier

Opis

Three coherent parts form the material covered in this text, portions of which have not been widely covered in traditional textbooks. In this coverage the reader is quickly introduced to several different topics enriched with 175 exercises which focus on real-world problems. Exercises range from the classics of probability theory to more exotic research-oriented problems based on numerical simulations. Intended for graduate students in mathematics and applied sciences, the text provides the tools and training needed to write and use programs for research purposes. The first part of the text begins with a brief review of measure theory and revisits the main concepts of probability theory, from random variables to the standard limit theorems. The second part covers traditional material on stochastic processes, including martingales, discrete-time Markov chains, Poisson processes, and continuous-time Markov chains. The theory developed is illustrated by a variety of examples surrounding applications such as the gambler's ruin chain, branching processes, symmetric random walks, and queueing systems. The third, more research-oriented part of the text, discusses special stochastic processes of interest in physics, biology, and sociology. Additional emphasis is placed on minimal models that have been used historically to develop new mathematical techniques in the field of stochastic processes: the logistic growth process, the Wright -Fisher model, Kingman's coalescent, percolation models, the contact process, and the voter model. Further treatment of the material explains how these special processes are connected to each other from a modeling perspective as well as their simulation capabilities in C and Matlab(TM).

Recenzje

"Stochastic Modeling by Nicolas Lanchier is an introduction to stochastic processes accessible to advanced students and interdisciplinary scientists with a background in graduate-level real analysis. The work offers a rigorous approach to stochastic models used in social, biological and physical sciences ... . Stochastic modeling provides a link between applied research in stochastic models and the literature covering their mathematical foundations."  (Ben Dyhr, Mathematical Reviews, May, 2018)

"There is a wide spectrum of topics discussed in this book. ... It is also interesting to find several classical examples with all details. ... The text is so carefully written and checked, that I was unable to find a single typo. The book can be strongly recommended to those students and teachers who want to be in line with modern probability theory and its diverse applications." (Jordan M. Stoyanov, zbMATH 1360.60002, 2017)

Spis treści

1. Basics of Measure and Probability Theory.- 2. Distribution and Conditional Expectation.- 3. Limit Theorems.- 4. Stochastic Processes: General Definition.- 5. Martingales.- 6. Branching Processes.- 7. Discrete-time Markov Chains.- 8. Symmetric Simple Random Walks.- 9. Poisson Point and Poisson Processes.- 10. Continuous-time Markov Chains.- 11. Logistic Growth Process.- 12. Wright-Fisher and Moran Models.- 13. Percolation Models.- 14. Interacting Particle Systems.- 15. The Contact Process.- 16. The Voter Model.- 17. Numerical Simulations in C and Matlab.

O autorze

Nicolas Lanchier is Associate Professor at Arizona State University, School of Mathematical and Statistical Sciences.  His research interests include mathematical models introduced in the life and social sciences literature that describe inherently spatial phenomena of interacting populations consist of systems of ordinary differential equations.  

Blurb

Three coherent parts form the material covered in this text, portions of which have not been widely covered in traditional textbooks. In this coverage the reader is quickly introduced to several different topics enriched with 175 exercises which focus on real-world problems. Exercises range from the classics of probability theory to more exotic research-oriented problems based on numerical simulations. Intended for graduate students in mathematics and applied sciences, the text provides the tools and training needed to write and use programs for research purposes.

The first part of the text begins with a brief review of measure theory and revisits the main concepts of probability theory, from random variables to the standard limit theorems. The second part covers traditional material on stochastic processes, including martingales, discrete-time Markov chains, Poisson processes, and continuous-time Markov chains. The theory developed is illustrated by a variety of examples surrounding applications such as the gambler’s ruin chain, branching processes, symmetric random walks, and queueing systems. The third, more research-oriented part of the text, discusses special stochastic processes of interest in physics, biology, and sociology. Additional emphasis is placed on minimal models that have been used historically to develop new mathematical techniques in the field of stochastic processes: the logistic growth process, the Wright–Fisher model, Kingman’s coalescent, percolation models, the contact process, and the voter model. Further treatment of the material explains how these special processes are connected to each other from a modeling perspective as well as their simulation capabilities in C and Matlab™.